Analysis of Stochastic State-Dependent Arrivals in a Queueing-Inventory System with Multiple Server Vacation and Retrial Facility

Abstract

This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of ${\lambda }_1\left({\theta }_1\right)$. If the server is not on vacation, the primary (retrial) arrival occurs with a rate of ${\lambda }_2\left({\theta }_2\right)$. Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of $\mu$, which follows exponential distribution. When the inventory level drops to a fixed s, reorder of Q items is triggered immediately under $(s,Q)$ ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.

1. Introduction

The analysis of vacation in a queueing-inventory system is being widely discussed nowadays. Psychologists have also suggested that providing tea or snack break to the employers during their work hours reduces their anxiety in work and tends to improve performance and efficiency in their work. The situation of a server taking a vacation or break is more realistic compared with other models in queueing-inventory models.

1.1. Motivation

The following real-life situation inspired the author to bring out this stochastic modelling in the vacation queueing-inventory system. Recently, the author went to a small textile showroom to purchase a shirt. It runs with a single server and a finite waiting hall. When the server took vacation, the author entered the showroom and joined the queue. He realised that the average arrival rate of a customer was becoming very low. A few minutes later, that is, after the server turns back to the regular service mode, the average arrival rate of a customer is higher than during the server’s vacation period. The author thought about these two average arrival rates in the single server textile shop according to the availability of the server. Finally, he decided to implement these server-state dependent arrival rates in the stochastic queueing-inventory modelling.

1.2. Contribution of the Model

The following assumptions are to be carried out in this paper.

1.

This model analyses the state-dependent arrival pattern on the expected system performance measures.

2.

This model brings out the stability condition and the stationary probability vector of the proposed system using the Neuts matrix analytic method.

3.

Using numerical simulation, it investigates the expected total cost of the model under parameter variation.

4.

The fraction of time of a server’s vacation is to be discussed in detail with a numerical illustration.

5.

The customer’s waiting time, the fraction of successful rate of retrial, the expected mean inventory level, the number of customers in the waiting hall, thes and orbit are to be analysed using the parameter variation.

1.3. Literature Review

In the earlier period of research into queueing-inventory systems (QIS), it was assumed that when a customer reached the server, the providing service would be instantaneous and the item would be delivered to the customer instantly. However, it is not necessarily true in a real-life phenomenon. For example, readers can see it in a computer showroom when a customer purchases a product. The salesman explains the features of the product and the advantages of purchasing the product, which takes some random period of time generally. Thus, the need to introduce the concept of positive service time in QIS was required to do the service process. Initially, positive service time was introduced in the queueing system by Melikov [1] and Sigman and Simchi-Levi [2]. Berman et al. [3] framed a dynamical problem with a deterministic service time. For more details regarding the positive service time QIS, one can refer to [4,5,6,7,8,9].

State-dependent arrival and service rates on the queue were examined by Gupta [10]. He predicted an exponential distribution for the service and a hyper-Poisson distribution for the arrival. Boxma et al. [11] considered a storage model with state-dependent arrival. The content will increase at some state-dependent rate if the production is on, and the system will switch to the off state. Similarly, when the system is off, content will be decreased at some state-dependent rate, and finally the system will be in the off state. Charan Jeet Singh et al. [12] investigated the state-dependent arrival and vacation on an $M/G/1$ queueing model.

Jeganathan et al. [13] considered a retrial queueing inventory system with stock-dependent arrival and compared the ordering policies $(s,Q)$ and $(s,S)$ in which the $(s,S)$ ordering policy is more efficient by the numerical example. Abdul Reiyas and Jeganathan [14] investigated a state-dependent arrival queueing-inventory model which is based on the present inventory in the system. Nazarov et al. [15] derived pseudo steady state period for the infinite server queue, whereas the customer arrival rate dependent on the state. Here, the customer arrival rate is dependent on the current number of customers in the system. For more on the state-dependent arrival, refer to the following: [16,17,18,19,20].

During the stock out period or a server’s vacation period, any arriving customer’s demand will not be fulfilled. Those customers may leave the system for a certain time to have a cup of coffee or to do some other work in another place and later join the queue. This retrial concept plays an important role in the field of queuing-inventory theory. At the beginning, the retrial concept with an inventory system was studied by Artaljeo et al. [21]. Paul Manuel et al. [22] analysed a perishable inventory system with retrial and negative demands. Usually, the retrial of any individual customer does not depend on the other customers in orbit. However, the rate of retrial customers is directly proportional to the number of customers in orbit, which is referred to as the classical retrial inventory system (CRIS). Ushakumari [23] established an analytical approach to CRIS. Jeganathan et al. [24] analysed a retrial queueing-inventory system with discretionary priority. Recently, in the single server queueing-inventory system, Nithya et al. [25] analysed the classical retrial facility with two commodities.

When the server cannot provide its subsequent service, the server is permitted to take a vacation in the queueing-inventory model. In the system, the server takes a rest for a random time once the server’s busy period comes to an end. In general, taking a break allows the server to re-energize and continue providing services without getting worn out. The likelihood of a server finishing his busy period is as follows when observing the QIS: (1) when there is no stock available; (2) when there are no customers in the system. Similar to that, the QIS also provides information about vacation return rules. Upon completion of vacation, the server checks the queue and inventory level. If the server finds any symptoms of the server becoming busy, he will come to the system immediately and start service to the customers; otherwise he will take another vacation, which is called a multiple vacation. In addition, there are more types of vacation that have been studied by many researchers in the field of QIS, such as working vacation, delayed vacation, differentiated vacation, and so on. However, in this paper, we concentrate on the servers’ multiple vacations in the stochastic QIS.

The vacation idea in a QIS was introduced by Daniel and Ramanarayanan [26]. They used a term rest period to represent the vacation period of a server. They allowed the server to take a rest when the number of items in the inventory became zero and assumed that the duration of vacation time is generally distributed in [27]. Customers who arrive in the system when the server performs a vacation task are considered to be lost. Recently, Yuying Zhang et al. [28] considered a multiple vacation of a server in the QIS. They applied a random order size policy (ROS) for the replenishment process and lost sales concept during the arrival epoch at which the server is on vacation. Dequan Yue and Yaling Qin [29] explored the production QIS in which they assumed that the production facility will take vacation if the current stock level is reached to S items. Similarly, it ends the vacation whenever the number of items in the inventory drops to s. Koroliuk et al. [30] investigated the vacation task of the server in the QIS. One can notice that they allow the server to take a vacation if there is no customer in the queue and allow him to complete his vacation whenever he finds that the queue size has reached the fixed threshold level. Otherwise, his vacation period is to be renewed.

Yadavalli and Jeganathan [31] explored a perishable retrial inventory system with a finite queue and two heterogeneous servers, one of which can take multiple vacations. Considering a finite retrial inventory system with server vacation, Jeganathan et al. [32] studied the significance of heterogeneous servers over homogeneous servers, and they have shown that heterogeneous servers are the most effective. Narayanan et al. [33] considered an inventory system with a positive service time in which the arrival pattern of a customer is determined according to a Markovian arrival process and the service time of a customer follows a phase-type distribution. The customers waiting for their respective services may renege randomly. The server can also take multiple vacations if there are no customers waiting in the system, or if there is no stock available in the inventory. Kathiresan et al. [34] considered a single server retrial queueing-inventory system with server vacation. Here, the server can take multiple vacations and provide two different types of services to the arriving customer. Furthermore, Revathi and Francis Raj [35] investigated a single-server retrial queuing system with a modified Bernoulli vacation. One is primary service, in which an item would be deducted, and the other one is service alone. Sugapriya et al. [36] analysed an inventory system without a service facility in which customer arrival depends on the number of items and the server goes on vacation when the inventory is empty.

Sivakumar [37] analysed a retrial inventory system with multiple server vacations. The server will take a vacation if the inventory is zero. After completing the vacation, if there is no item available in the inventory, the server will continue the vacation, which is known as a multiple vacation. Manikandan and Nair Sanjeev [38] discussed an $M/M/1/1$ queueing-inventory system where customers may enter into the orbit with unsatisfied service. Jeganathan et al. [39] studied a single server retrial queueing inventory system with a queue-dependent service rate. Recently, Jeganathan et al. [40] discussed the two types of inventory with a queue-dependent service facility in a retrial queueing-inventory system. In the recent times, Radhamani [41] has made a comparative study on the three different reordering policies, which are variable-size order policy, fixed-size order policy, and order up to S policy in the QIS under the multiple server vacation. The results show that the optimum total cost occurs under the variable-size ordering policy.

Poonam Gupta et al. [42] considered a queueing system with impatient customers and provided two kinds of vacation options, such as classical vacation and working vacation, after all the customers got service under Bernoulli’s trial. Zhang et al. [43] analysed a queuing inventory model with server multiple vacation and impatient customers under the $(s,S)$ ordering policy. The arrival customer may wait in the queue for their service with probability p or lost with the probability of $1-p$. Krishnamoorthy et al. [44] analysed a queueing inventory system where the server takes a vacation in the Markovian arrival process. The server can start the vacation when a service process is stopped until a minimum number of customers gather to initiate the next batch service. While the server is engaged on vacation, production of items is started until the inventory reaches a certain level or until the number of customers in the system reaches a maximum service batch size. Soujanya et al. [45] analysed a queueing-inventory model with server vacation, where the server starts vacation whenever the queue becomes empty. In a multi-server queueing system, Bouhentouf et al. [46] studied the vacation and impatience of customers. A server is allowed to take a vacation when the queue becomes empty.

Beena and Jose [47] studied the production inventory system with multiple server vacations. The servers provide service only when there are customers in the waiting area and the stock level is positive; otherwise they start vacations. If there are no customers or items in the inventory or both, it again chooses vacation. The server can continue vacations until they find at least one member in the system with a positive stock level. Ayyappan et al. [48] analysed a queueing system with server breakdown and instantaneous feedback customers. When the system has no customers, the server can take a random vacation, and the server will wait for the customer’s arrival after returning from the vacation. Manikandan and Nair [49] analysed the working vacation and interruption in the queueing-inventory system. Instead of taking a break, servers will be working at a slow rate during the working vacation. The server begins vacation whenever the inventory or the waiting hall becomes empty. An uncertainty circumstance has been discussed by Qu et al. [50]. The model having risk-averse two-stage stochastic minimum cost models has been analysed by Ji et al. [51].

1.4. Research Gap

The following assumptions are found as a research gap in the stochastic queueing-inventory literature, regarding the state-dependent arrival process with respect to the server status in an infinite retrial multiple server vacation queueing-inventory system. No paper has discussed the vacation-dependent arrival rate for both primary and retrial customers. The vacation offer of the server is utilised only when both no positive stock and no positive queue appear. However, in the literature, Sivakumar [37] assumed that if there is no positive stock, then the server goes on vacation. The vacation assumption of our model is also a new attempt.

1.5. Proposal of the Model

The stochastic model is developed using the following assumptions:

1.

A server status-dependent arrival rate is assumed for both primary and retrial arrivals in a vacation queueing-inventory system.

2.

The server vacation offer is utilised only when both zero stock and zero customers in the queue occur.

3.

Multiple server vacation policy is assumed.

4.

In addition, the $(s,Q)$ ordering principle is adopted for the replenishment of the inventory.

5.

If the waiting hall is full, an arriving new customer sends to an infinite orbit or leaves the system under the Bernoulli schedule.

2. Model Definition

This model considers a single server queueing-inventory system with multiple server vacations and a state-dependent arrival process. In this system, the server can do two different tasks as follows: (1) the server is on vacation task; (2) the server is on regular (not on vacation) task.

Definition 1. 

Vacation policy: The server is allowed to take a vacation whenever there is no customer in the waiting hall and no item available in the inventory. Otherwise, the server stays in the system to do regular tasks (not on vacation). After vacation completion, the server checks both the waiting hall and inventory. If the server identifies that both are zero, he/she takes another vacation. Whenever the server faces the same situation at the time of coming back from vacation, he/she continuously takes another vacation. This vacation is called multiple server vacation.

Definition 2. 

State-dependent arrival pattern for primary arrival: The system classifies the arrival pattern of the customer depend upon the task of the server. If the server is on vacation task, the arriving customer has the arrival rate ${\lambda }_1$ and if the server is not on a vacation task, the arriving customer has the arrival rate ${\lambda }_2$. The arrival flow in each state of the server follows an independent Poisson process. The system assumes that ${\lambda }_1\le {\lambda }_2$ and both ${\lambda }_1>0$ and ${\lambda }_2>0$.

Definition 3. 

State-dependent arrival pattern for retrial arrival: The retrial arrival is also assumed to be state-dependent as in Definition 2. Thus, ${\theta }_1$ and ${\theta }_2$ are the arrival rates of retrial customers when the server is on vacation task and not on vacation task, respectively. The inter-retrial time between two successive retrial arrivals in each task of the server follows an exponential distribution. Here, it is be noted that ${\theta }_1\le {\theta }_2$ and ${\theta }_1>0$, ${\theta }_2>0.$

2.1. Model Explanation

The system will have a maximum of S units of inventory, N seats in the finite waiting hall, infinite orbit capacity, and a single server. As we said earlier, the server can either be engaged in a vacation task or not on a vacation task. While the server is not on a vacation task, it can either be busy or idle. If the server finds that there is at least one item in the inventory and one customer in the queue, then only he/she will be busy. In either case of positive inventory and zero customers in the queue or zero items in a positive queue, the server will be idle. An arriving customer can obtain service immediately if the system has positive inventory and the server is idle during the regular task. Otherwise, he/she must wait in the waiting hall. If the waiting hall starts to overflow, an arriving customer can be directed to orbit with their corresponding arrival rate. The arrival process is executed as in Definition 2.

The customer in orbit can retry to enter the system at any time. However, the retrial process will be successful only if there are fewer than N seats available in the waiting hall. Their retrial process of orbit customers is done as we defined in Definition 3. Whenever the server observes a positive customer in the waiting hall and positive stock, he/she shall start service to the customer following the first come, first serve discipline. The service rate per customer is defined as $\mu$ ($\mu >0$). The average service time per customer follows an exponential distribution. When the time of service completion of the last customer and the last item in the inventory arrives, the server goes for a vacation task according to Definition 1.

The server’s vacation completion rate is denoted as $\gamma$. The average vacation time follows an exponential distribution. The server will come back to the regular task if the duration of the vacation is over and there will be either a positive inventory or a positive number of customers in the waiting hall. Otherwise, the server remains on the vacation task. The replenishment process of this system is to be executed by following the $(s,Q)$ ordering principle. The rate $\beta$ indicates that the replenishment is done by the supplier. The average reorder time is assumed to be exponentially distributed.

2.2. Matrix Formulation

The collection $X\left(t\right)=\{(X_1\left(t\right),X_2\left(t\right),X_3\left(t\right),X_4\left(t\right):t\ge 0\}$ constitutes a four-dimensional stochastic process with the state space $E={\displaystyle \bigcup _{x=1}^3}{E}_x$, where

$X_1\left(t\right)$—number of customers present in the orbit at time t

$X_2\left(t\right)$—number of items available at the inventory at time t

$X_3\left(t\right)$—the status of server: $\left\{\begin{array}{cc}0\hfill & \mathrm{if}\mathrm{server}\mathrm{is}\mathrm{on}\mathrm{vacation}\hfill \\ 1\hfill & \mathrm{if}\mathrm{server}\mathrm{is}\mathrm{not}\mathrm{on}\mathrm{vacation}\hfill \end{array}\right.$

$X_4\left(t\right)$—the number of customers present in the waiting hall at time t and

$E_1=\left\{({\alpha }_1,{\alpha }_2,0,{\alpha }_3):{\alpha }_1=0,1,2,\dots ;{\alpha }_2=0,Q;{\alpha }_3=0,1,2,\dots N\right\}$

$E_2=\left\{({\alpha }_1,{\alpha }_2,1,{\alpha }_3):{\alpha }_1=0,1,2,\dots ;{\alpha }_2=0;{\alpha }_3=1,2,\dots N\right\}$

$E_3=\left\{({\alpha }_1,{\alpha }_2,1,{\alpha }_3):{\alpha }_1=0,1,2,\dots ;{\alpha }_2=1,2,\dots ,S;{\alpha }_3=0,1,2,\dots N\right\}$

Since the state space is discrete, $X\left(t\right)$ holds the continuous-time Markov chain property. Furthermore, as all the states in E are reachable from every other state, we say that $X\left(t\right)$ is an irreducible Markov chain. This $X\left(t\right)$ forms a quasi-birth and death process. It can be easily seen from the structure of an infinitesimal matrix P,

$$P=\begin{array}{cc}& \begin{array}{cccccccc}0& 1& 2& \dots & & & & \end{array}\\ \begin{array}{c}0\\ 1\\ 2\\ ⋮\end{array}& \left(\begin{array}{ccccccc}{A}_0& C& & & & & \\ A_2& A_1& C& & & & \\ & A_2& A_1& C& & & \\ & \ddots & \ddots & \ddots & & \end{array}\right)\end{array}$$

where

$${\left[C\right]}_{[(S+2)(N+1)+N]}=\begin{array}{cc}& \begin{array}{ccccccc}(0,0)& (Q,0)& (0,1)& (1,1)& \dots & (S-1,1)& (\mathrm{S},1)\end{array}\\ \begin{array}{c}(0,0)\\ (Q,0)\\ (0,1)\\ (1,1)\\ ⋮\\ ⋮\\ (\mathrm{S},1)\end{array}& \left(\begin{array}{ccccccc}{C}_0\hspace{1em}& & & & & & \\ & & C_0\hspace{1em}& & & & \\ & & & C_1\hspace{1em}& & & \\ & & & & C_2\hspace{1em}& & \\ & & & & & \ddots \hspace{1em}& \\ & & & & & & C_2\hspace{1em}& & \\ & & & & & & & C_2\end{array}\right)\end{array}$$

$$C_{0_{[N+1]}}=\left\{\begin{array}{cc}{\lambda }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$C_{1_{\left[N\right]}}=\left\{\begin{array}{cc}{\lambda }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$C_{2_{[N+1]}}=\left\{\begin{array}{cc}{\lambda }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[A_2\right]}_{\left[\right(S+2\left)\right(N+1)+N]}=\begin{array}{cc}& \begin{array}{ccccccc}(0,0)& (Q,0)& (0,1)& (1,1)& \dots & (S-1,1)& (S,1)\end{array}\\ \begin{array}{c}(0,0)\\ (Q,0)\\ (0,1)\\ (1,1)\\ ⋮\\ ⋮\\ (S,1)\end{array}\left(\begin{array}{ccccccc}{B}_0\hspace{1em}& & & & & & \\ & B_0\hspace{1em}& & & & & \\ & & B_1\hspace{1em}& & & & \\ & & & B_2\hspace{1em}& & & \\ & & & & \ddots \hspace{1em}& & \\ & & & & & B_2\hspace{1em}& \\ & & & & & & B_2\end{array}\right)\end{array}$$

$$B_{0_{[N+1]}}=\left\{\begin{array}{cc}{\theta }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0,1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$B_{1_{\left[N\right]}}=\left\{\begin{array}{cc}{\theta }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$B_{2_{[N+1]}}=\left\{\begin{array}{cc}{\theta }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0,1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

For $i=0,1.$

$${\left[A_i\right]}_{\left[\right(S+2\left)\right(N+1)+N]}=\begin{array}{cc}& \begin{array}{cc}0& 1\end{array}\\ \begin{array}{c}0\\ 1\end{array}\left(\begin{array}{cc}{B}_{i0}& A_{01}\\ A_{10}& B_{i1}\end{array}\right)\end{array}$$

where

$${\left[B_{i0}\right]}_{2\left[\right(N+1\left)\right]}=\begin{array}{cc}& \begin{array}{cc}(0,0)& (Q,0)\end{array}\\ \begin{array}{c}(0,0)\\ (Q,0)\end{array}\left(\begin{array}{cc}{D}_{i0}^{\left(0\right)}& F_1\\ 0& D_{iQ}^{\left(0\right)}\end{array}\right)\end{array}$$

$${\left[A_{01}\right]}_{\left[2\right(N+1)\times N+S(N+1\left)\right]}=\begin{array}{cc}& \begin{array}{ccccccc}(0,1)& (1,1)& \dots & (Q,1)& (Q+1,1)& \dots & (S,1)\end{array}\\ \begin{array}{c}(0,0)\\ (Q,0)\end{array}\left(\begin{array}{ccccccc}{F}_2\hspace{1em}& 0\hspace{1em}& \dots \hspace{1em}& 0\hspace{1em}& 0\hspace{1em}& \dots \hspace{1em}& 0\\ 0\hspace{1em}& 0\hspace{1em}& \dots \hspace{1em}& F_3\hspace{1em}& 0\hspace{1em}& \dots \hspace{1em}& 0\end{array}\right)\end{array}$$

$${\left[A_{10}\right]}_{[N+S(N+1)\times 2(N+1\left)\right]}=\begin{array}{cc}& \begin{array}{cc}(0,1)& (Q,1)\end{array}\\ \begin{array}{c}(0,1)\\ (1,1)\\ (2,1)\\ ⋮\\ (S,1)\end{array}\left(\begin{array}{cc}0\hspace{1em}& 0\\ F_3\hspace{1em}& 0\hspace{1em}\\ 0\hspace{1em}& 0\hspace{1em}\\ ⋮\hspace{1em}& ⋮\\ 0\hspace{1em}& 0\end{array}\right)\end{array}$$

$${\left[B_{i1}\right]}_{[N+S(N+1\left)\right]}=\left\{\begin{array}{cc}{D}_{ij}^{\left(1\right)},\hfill & {\alpha }_2^{\prime }={\alpha }_2,{\alpha }_2=0,1,\dots ,S\hfill \\ F_5,\hfill & {\alpha }_2^{\prime }={\alpha }_2-1,{\alpha }_2=1\hfill \\ F_6,\hfill & {\alpha }_2^{\prime }={\alpha }_2-1,{\alpha }_2=2,3,\dots ,S\hfill \\ F_7,\hfill & {\alpha }_2^{\prime }={\alpha }_2+Q,{\alpha }_2=0\hfill \\ F_8,\hfill & {\alpha }_2^{\prime }={\alpha }_2+Q,{\alpha }_2=1,2,\dots ,s\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

where

$${\left[D_{i0}^{\left(0\right)}\right]}_{[N+1]}=\left\{\begin{array}{cc}-({\lambda }_1+\beta +{\theta }_1),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0\hfill \\ -({\lambda }_1+\beta +{\theta }_1+\gamma ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -({\lambda }_1+\beta +\gamma ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$F_1[N+1]=\beta I_{[N+1]}$$

$${\left[F_2\right]}_{\left[\right(N+1)\times N]}=\left\{\begin{array}{cc}\gamma ,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$F_3[N+1]=\gamma I_{[N+1]}$$

$${\left[D_{iQ}^{\left(0\right)}\right]}_{N+1}=\left\{\begin{array}{cc}-({\lambda }_1+{\theta }_1+\gamma ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0,1,\dots ,N-1\hfill \\ -({\lambda }_1+\gamma ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[F_4\right]}_{[N+1]}=\left\{\begin{array}{cc}\mu ,\hfill & {\alpha }_3^{\prime }={\alpha }_3-1,{\alpha }_3=1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[F_5\right]}_{[N+1]}=\left\{\begin{array}{cc}\mu ,\hfill & {\alpha }_3^{\prime }={\alpha }_3-1,{\alpha }_3=2.3,\dots ,N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[F_6\right]}_{[N+1]}=\left\{\begin{array}{cc}\mu ,\hfill & {\alpha }_3^{\prime }={\alpha }_3-1,{\alpha }_3=1,2,\dots ,N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[F_7\right]}_{[N\times (N+1\left)\right]}=\left\{\begin{array}{cc}\beta ,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[F_8\right]}_{[N+1]}=\beta I_{[N+1]}$$

$${\left[D_{i0}^{\left(1\right)}\right]}_{\left[N\right]}=\left\{\begin{array}{cc}-({\lambda }_2+\beta +{\theta }_2),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -(\beta +{\lambda }_2),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

For $j=1,2,\dots ,S-1$

$${\left[D_{ij}^{\left(1\right)}\right]}_{[N+1]}=\left\{\begin{array}{cc}-({\lambda }_2+{\theta }_2+\beta ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0\hfill \\ -({\lambda }_2+{\theta }_2+\beta +\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -({\lambda }_2+\beta +\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$${\left[D_{iS}^{\left(1\right)}\right]}_{[N+1]}=\left\{\begin{array}{cc}-({\lambda }_2+{\theta }_2),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0\hfill \\ -({\lambda }_2+{\theta }_2+\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -({\lambda }_2+\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

3. Main Results

Here, the necessary and sufficient condition of the proposed system’s stability is to be derived. On following that, we will find an $R-$ matrix and stationary probability vector to the system.

3.1. Stability Analysis

Since $X\left(t\right)$ forms a infinite dimensional QBD process, we need to find the stability condition of the proposed model. It is to be derived here with the help of the Neuts matrix analytic method [52]. To proceed further, we consider the matrix $\tilde{A}$ where $\tilde{A}=A_2+A_1+C$, which is given by

$$\tilde{A}=\begin{array}{cc}& \begin{array}{cc}0& 1\end{array}\\ \begin{array}{c}0\\ 1\end{array}\left(\begin{array}{cc}{\tilde{A}}_0& {\tilde{A}}_{01}\\ {\tilde{A}}_{10}& {\tilde{A}}_1\end{array}\right)\end{array}$$

where

$${\tilde{A}}_{10}=\begin{array}{cc}& \begin{array}{cc}(0,0)& (Q,0)\end{array}\\ \begin{array}{c}(0,0)\\ (Q,0)\end{array}\left(\begin{array}{cc}{A}_0^{\left(0\right)}& F_1\\ 0& A_Q^{\left(0\right)}\end{array}\right)\end{array}$$

$$\widetilde{A_1}=\left\{\begin{array}{cc}{A}_0^{\left(1\right)},\hfill & {\alpha }_2^{\prime }={\alpha }_2,{\alpha }_2=0\hfill \\ A,\hfill & {\alpha }_2^{\prime }={\alpha }_2,{\alpha }_2=1,2,\dots ,s\hfill \\ B,\hfill & {\alpha }_2^{\prime }={\alpha }_2,{\alpha }_2=s+1,s+2,\dots ,S\hfill \\ F_5,\hfill & {\alpha }_2^{\prime }={\alpha }_2-1,{\alpha }_2=1\hfill \\ F_6,\hfill & {\alpha }_2^{\prime }={\alpha }_2-1,{\alpha }_2=2,3,\dots ,S\hfill \\ F_7,\hfill & {\alpha }_2^{\prime }={\alpha }_2+Q,{\alpha }_2=0\hfill \\ F_8,\hfill & {\alpha }_2^{\prime }={\alpha }_2+Q,{\alpha }_2=1,2,\dots ,s\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$A_0^{\left(0\right)}=\left\{\begin{array}{cc}-({\lambda }_1+\beta +{\theta }_1),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0\hfill \\ -({\lambda }_1+\beta +{\theta }_1+\gamma ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -(\beta +{\theta }_1+\gamma ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_1+{\theta }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$A_Q^{\left(0\right)}=\left\{\begin{array}{cc}-({\lambda }_1+\gamma +{\theta }_1+),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -{\lambda }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_1+{\theta }_1,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$A_0^{\left(1\right)}=\left\{\begin{array}{cc}-({\lambda }_2+\beta +{\theta }_2),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -\beta ,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_2+{\theta }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=1,2,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$A=\left\{\begin{array}{cc}-({\lambda }_2+{\theta }_2),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0\hfill \\ -({\lambda }_2+{\theta }_2+\beta +\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -(\beta +\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_2+{\theta }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$B=\left\{\begin{array}{cc}-({\lambda }_2+{\theta }_2),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=0\hfill \\ -({\lambda }_2+{\theta }_2+\mu ),\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=1,2,\dots ,N-1\hfill \\ -\mu ,\hfill & {\alpha }_3^{\prime }={\alpha }_3,{\alpha }_3=N\hfill \\ {\lambda }_2+{\theta }_2,\hfill & {\alpha }_3^{\prime }={\alpha }_3+1,{\alpha }_3=0,1,\dots ,N-1\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

Lemma 1. 

The steady state probability vector $\mathsf{\Pi }=({\pi }^{(0,0)}{\pi }^{(Q,0)}{\pi }^{(0,1)}{\pi }^{(1,1)}{\pi }^{(2,1)}\dots {\pi }^{(S,1)})$ to the generator matrix $\tilde{A}$ is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\pi }^{({\alpha }_2,1)}& ={\pi }^{(Q,1)}{\mathsf{\Omega }}_{({\alpha }_2,1)}\forall {\alpha }_2=0,1,2,\dots ,S\hfill \\ \hfill and\\ \hfill {\pi }^{({\alpha }_2,0)}& ={\pi }^{(Q,1)}{\mathsf{\Omega }}_{({\alpha }_2,0)}\forall {\alpha }_2=0,Q,\hfill \end{array}\end{array}$$

(1)

where

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_{({\alpha }_2,1)}& =& \left\{\begin{array}{cc}{(-1)}^Q{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}[F_5-F_4{A}_0^{\left(0\right)-1}{F}_2]F^{-1},\hfill & {\alpha }_2=0\hfill \\ {(-1)}^{Q-{\alpha }_2}{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}{\left(F_6{A}^{-1}\right)}^{s+1-{\alpha }_2},\hfill & {\alpha }_2=1,2,\dots ,s\hfill \\ {(-1)}^{Q-{\alpha }_2}{\left(F_6{B}^{-1}\right)}^{Q-(s+1)},\hfill & {\alpha }_2=s+1,s+2,\dots ,Q-1\hfill \\ I,\hfill & {\alpha }_2=Q\hfill \\ {(-1)}^{Q-[{\alpha }_2-(Q+1)]}{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}\left(F_6{A}^{-1}\right)\hfill \\ \{\sum _{l=0}^{S-({\alpha }_2-1)}{\left(F_8{B}^{-1}\right)}^2{\left(F_6{B}^{-1}\right)}^{S-({\alpha }_2-1)}+\hfill \\ {\left(F_6{A}^{-1}\right)}^{S-({\alpha }_2-l)}\left(F_8{B}^{-1}\right){\left(F_6{B}^{-1}\right)}^l\},\hfill & {\alpha }_2=Q+1,Q+2,\dots ,S-2\hfill \\ {(-1)}^{Q-s+2}{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}\left(F_6{A}^{-1}\right)[F_6{A}^{-1}+\hfill \\ F_8{B}^{-1}]F_8{B}^{-1},\hfill & {\alpha }_2=S-1\hfill \\ {(-1)}^{Q-s+1}{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}\left(F_6{A}^{-1}\right)F_8{A}^{-1}\hfill & {\alpha }_2=S\hfill \end{array}\right.\hfill \\ \hfill {\mathsf{\Omega }}_{({\alpha }_2,0)}& =& \left\{\begin{array}{cc}{(-1)}^Q{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}{\left(F_6{A}^{-1}\right)}^s{F}_4{A}_0^{{\left(0\right)}^{-1}},\hfill & {\alpha }_2=0\hfill \\ {(-1)}^{Q+1}{\left(F_6{B}^{-1}\right)}^{Q-(s+1)}{\left(F_6{A}^{-1}\right)}^s{F}_4{A}_0^{-1}{F}_1{A}_Q^{{\left(0\right)}^{-1}},\hfill & {\alpha }_2=Q\hfill \end{array}\right.\hfill \end{array}$$

and ${\pi }^{(Q,1)}$ can be obtained by substituting the values in $\mathsf{\Pi }\mathbf{e}=1.$

Proof. 

The steady state probability vector $\mathsf{\Pi }$ satisfies $\mathsf{\Pi }\tilde{A}=\mathbf{0},\mathsf{\Pi }\mathbf{e}=1$

$$\begin{array}{cc}\hfill {\pi }^{(0,0)}{A}_0^{\left(0\right)}+{\pi }^{(1,1)}{F}_4& =\mathbf{0},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\pi }^{(0,0)}{F}_1+{\pi }^{(Q,0)}{A}_Q^{\left(0\right)}& =\mathbf{0}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\pi }^{({\alpha }_2,0)}{F}_2+{\pi }^{({\alpha }_2,1)}{A}_0^{\left(1\right)}+{\pi }^{({\alpha }_2+1,1)}{F}_5& =\mathbf{0},{\alpha }_2=0\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\pi }^{({\alpha }_2,1)}A+{\pi }^{({\alpha }_2+1,1)}{F}_6& =\mathbf{0},{\alpha }_2=1,\dots s\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\pi }^{({\alpha }_2,1)}B+{\pi }^{({\alpha }_2+1,1)}{f}_6& =\mathbf{0},{\alpha }_2=s+1,\dots Q-1\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\pi }^{({\alpha }_2,0)}{F}_3+{\pi }^{(0,1)}{F}_7+{\pi }^{({\alpha }_2,1)}B+{\pi }^{({\alpha }_2+1,1)}{F}_6& =\mathbf{0},{\alpha }_2=Q\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\pi }^{({\alpha }_2-Q,1)}{F}_8+{\pi }^{({\alpha }_2,1)}B+{\pi }^{({\alpha }_2+1,1)}{F}_6& =\mathbf{0},{\alpha }_2=Q+1,\dots S-1\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\pi }^{({\alpha }_2-Q,1)}{F}_8+{\pi }^{({\alpha }_2,1)}B& =\mathbf{0},{\alpha }_2=S\hfill \end{array}$$

(9)

${\pi }^{(Q,1)}$ is obtained by solving (7) and using the normalising condition $\pi \mathbf{e}=1$.

$${\pi }^{(Q,1)}\left[{\mathsf{\Omega }}_{(Q,0)}{F}_3+{\mathsf{\Omega }}_{(0,1)}{F}_7+I+{\mathsf{\Omega }}_{(Q+1,1)}\right]=\mathbf{0}.$$

(10)

Solving (10) and

$${\pi }^{(Q,1)}\left[{\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,0)}+{\mathsf{\Omega }}_{(0,1)}+I+\sum _{{\alpha }_2=1,{\alpha }_2\ne Q}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}\right]=1,$$

we obtain ${\pi }^{(Q,1)}$. □

Lemma 2. 

The given queueing-inventory system is stable under the necessary condition

$$\begin{array}{c}\hfill \left({\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,0)}\right)C_0\mathbf{e}+{\mathsf{\Omega }}_{(0,1)}{C}_1\mathbf{e}+\sum _{{\alpha }_2=1}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}{C}_2\mathbf{e}>\left({\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,0)}\right)B_0\mathbf{e}+{\mathsf{\Omega }}_{(0,1)}{B}_1\mathbf{e}+\sum _{{\alpha }_2=1}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}{B}_2\mathbf{e}.\end{array}$$

(11)

Proof. 

Let us use the well-known result of Neuts $\mathsf{\Pi }{A}_2\mathbf{e}>\mathsf{\Pi }C\mathbf{e}$, we obtain

$$\begin{array}{cc}\hfill ({\pi }^{(0,0)}{\pi }^{(Q,0)}{\pi }^{(0,1)}{\pi }^{(1,1)}\dots {\pi }^{(S,1)})A_2\mathbf{e}& >({\pi }^{(0,0)}{\pi }^{(Q,0)}{\pi }^{(0,1)}{\pi }^{(1,1)}\dots {\pi }^{(S,1)})C\mathbf{e}\hfill \\ \hfill ({\pi }^{(0,0)}{B}_0+{\pi }^{(Q,0)}{B}_0+{\pi }^{(0,1)}{B}_1+{\pi }^{(1,1)}{B}_2+\dots +{\pi }^{(S,1)}{B}_2)\mathbf{e}& >({\pi }^{(0,0)}{C}_0+{\pi }^{(Q,0)}{C}_0+{\pi }^{(0,1)}{C}_1+{\pi }^{(1,1)}{C}_2+\dots +{\pi }^{(S,1)}{C}_2)\mathbf{e}\hfill \\ \hfill ({\pi }^{(0,0)}+{\pi }^{(Q,0)})B_0+{\pi }^{(0,1)}{B}_1+\sum _{{\alpha }_2=1}^S{\pi }^{({\alpha }_2,1)}{B}_2\mathbf{e}& >({\pi }^{(0,0)}+{\pi }^{(Q,0)})C_0+{\pi }^{(0,1)}{C}_1+\sum _{{\alpha }_2=1}^S{\pi }^{({\alpha }_2,1)}{C}_2\mathbf{e}\hfill \\ \hfill {\pi }^{(Q,1)}\left[({\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,1)})B_0+{\mathsf{\Omega }}_{(0,1)}{C}_1+\sum _{{\alpha }_2=1}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}{B}_2\right]\mathbf{e}& >{\pi }^{(Q,1)}[({\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,1)})C_0+{\mathsf{\Omega }}_{(0,1)}{C}_1+\sum _{{\alpha }_2=1}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}{C}_2]\mathbf{e}\hfill \\ \hfill \left({\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,0)}\right)B_0\mathbf{e}+{\mathsf{\Omega }}_{(0,1)}{B}_1\mathbf{e}+\sum _{{\alpha }_2=1}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}{B}_2\mathbf{e}& >\left({\mathsf{\Omega }}_{(0,0)}+{\mathsf{\Omega }}_{(Q,0)}\right)C_0\mathbf{e}+{\mathsf{\Omega }}_{(0,1)}{C}_1\mathbf{e}+\sum _{{\alpha }_2=1}^S{\mathsf{\Omega }}_{({\alpha }_2,1)}{C}_2\mathbf{e}.\hfill \end{array}$$

□

3.2. Steady State Analysis

From the structure of matrix P, and from Lemma 2, the Markov process $X\left(t\right)$ with the state space E is regular. Then there exists a stationary probability vector,

$\varphi =({\varphi }^{\left(0\right)},{\varphi }^{\left(1\right)},{\varphi }^{\left(2\right)},\dots )$, which satisfies the following condition:

$$\varphi P=\mathbf{0}\mathrm{and}{\displaystyle \sum _{i=0}^{\infty }}{\varphi }^{\left(i\right)}=1.$$

3.3. Computation of R-Matrix

Due to the special structure of matrix P and using $\varphi$, matrix R can be determined by

$$B_l{R}^2+A_{l}R+C=\mathbf{0}$$

(12)

where

$$R=\left(\begin{array}{ccccccc}{Z}^{1,1}& Z^{1,2}& Z^{1,3}& Z^{1,4}& Z^{1,5}& \dots & Z^{1,S+3}\\ Z^{2,1}& Z^{2,2}& Z^{2,3}& Z^{2,4}& Z^{2,5}& \dots & Z^{2,S+3}\\ Z^{3,1}& Z^{3,2}& Z^{3,3}& Z^{3,4}& Z^{3,5}& \dots & Z^{3,S+3}\\ Z^{4,1}& Z^{4,2}& Z^{4,3}& Z^{4,4}& Z^{4,5}& \dots & Z^{4,S+3}\\ ⋮& ⋮& ⋮& ⋮& ⋮& \dots & ⋮\\ Z^{S+3,1}& Z^{S+3,2}& Z^{S+3,3}& Z^{S+3,4}& Z^{S+3,5}& \dots & Z^{S+3,S+3}\end{array}\right)$$

where

$if{\alpha }_1=1,2,4,5,6,\dots ,S+3$ and ${\alpha }_2=1,2,4,5,6,\dots ,S+3$

$$Z^{{\alpha }_1,{\alpha }_2}={\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ r_0^{{\alpha }_1,{\alpha }_2}& r_1^{{\alpha }_1,{\alpha }_2}& r_2^{{\alpha }_1,{\alpha }_2}& \dots & r_N^{{\alpha }_1,{\alpha }_2}\end{array}\right)}_{(N+1)\times (N+1)}$$

and ${\alpha }_2=3$

$$Z^{{\alpha }_1,{\alpha }_2}={\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ r_1^{{\alpha }_1,{\alpha }_2}& r_2^{{\alpha }_1,{\alpha }_2}& r_3^{{\alpha }_1,{\alpha }_2}& \dots & r_N^{{\alpha }_1,{\alpha }_2}\end{array}\right)}_{(N+1)\times \left(N\right)}.$$

For ${\alpha }_1=3$

if ${\alpha }_2=1,2,4,5,6,\dots ,S+3$

$$Z^{{\alpha }_1,{\alpha }_2}={\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ r_0^{{\alpha }_1,{\alpha }_2}& r_1^{{\alpha }_1,{\alpha }_2}& r_2^{{\alpha }_1,{\alpha }_2}& \dots & r_N^{{\alpha }_1,{\alpha }_2}\end{array}\right)}_{\left(N\right)\times (N+1)}$$

and ${\alpha }_2=3$

$$Z^{{\alpha }_1,{\alpha }_2}={\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ r_1^{{\alpha }_1,{\alpha }_2}& r_2^{{\alpha }_1,{\alpha }_2}& r_3^{{\alpha }_1,{\alpha }_2}& \dots & r_N^{{\alpha }_1,{\alpha }_2}\end{array}\right)}_{\left(N\right)\times \left(N\right)}.$$

Let $s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}$ be the ${\alpha }_3^{th}$ column element in the last row of ${\alpha }_1{\alpha }_2^{th}$ block in the $R^2$ matrix.

For all ${\alpha }_1=1,2;4,5,\dots ,S+3,$

if ${\alpha }_2=1,$

$$\begin{array}{cc}\hfill -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\beta +{\theta }_1)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2}\mu & =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill {\theta }_1{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\beta +\gamma +{\theta }_1)& =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_1{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\beta +\gamma )+{\lambda }_1{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

and ${\alpha }_2=2,$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+{\theta }_1)+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta & =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill {\theta }_1{s}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\gamma +{\theta }_1)& =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_1{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\gamma )+{\lambda }_1{\delta }_{{\alpha }_1,{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

and ${\alpha }_2=3,$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-2}\gamma -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-2}\gamma +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2}\mu & =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2-2}\gamma +r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta )+{\lambda }_1{\delta }_{ij}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

and ${\alpha }_2=4,5,\dots ,S+3$

$$\begin{array}{cc}\hfill -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}& =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +\beta +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=s+4,\dots ,Q+2$

$$\begin{array}{cc}\hfill -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=Q+3$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-(Q+1)}\gamma -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+\gamma s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-(Q+1)}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+\\ \hfill r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu +r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta & =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}\beta +r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{i,j}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=Q+4,\dots ,S+2$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_3=S+3$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+{\theta }_2)& =0,\hspace{1em}{\alpha }_3=0\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)& =0,\hspace{1em}{\alpha }_3=1,2,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N.\hfill \end{array}$$

For ${\alpha }_1=3$

when ${\alpha }_2=1$

$$\begin{array}{cc}\hfill -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\beta +{\theta }_1)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_1{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\beta +\gamma +{\theta }_1)& =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_1{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\beta +\gamma )+{\lambda }_1{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=2,$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+{\theta }_1)+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_1{s}_{+}^{{\alpha }_1,{\alpha }_2}{r}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\gamma +{\theta }_1)& =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_1{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-1}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_1-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_1+\gamma )+{\lambda }_1{\delta }_{{\alpha }_1,{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=3,$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-2}\gamma -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-2}\gamma +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2}\mu & =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2-2}\gamma +r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta )+{\lambda }_1{\delta }_{ij}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=4,5,\dots ,s+3$

$$\begin{array}{cc}\hfill -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}& =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +\beta +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=s+4,\dots ,Q+2$

$$\begin{array}{cc}\hfill -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=Q+3$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-(Q+1)}\gamma -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+\beta +{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+\gamma s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-(Q+1)}+r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+\\ \hfill r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu +r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta & =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}\beta +r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{i,j}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_2=Q+4,\dots ,S+2$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+r_{{\alpha }_3+1}^{{\alpha }_1,{\alpha }_2+1}\mu & =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N\hfill \end{array}$$

when ${\alpha }_3=S+3$

$$\begin{array}{cc}\hfill r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta -r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}({\lambda }_2+{\theta }_2)& =0,\hspace{1em}{\alpha }_3=1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2+{\theta }_2)& =0,\hspace{1em}{\alpha }_3=2,3,\dots ,N-1\hfill \\ \hfill {\theta }_2{s}_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}+s_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2-Q}\beta +r_{{\alpha }_3-1}^{{\alpha }_1,{\alpha }_2}{\lambda }_2-r_{{\alpha }_3}^{{\alpha }_1,{\alpha }_2}(\mu +{\lambda }_2)+{\lambda }_2{\delta }_{{\alpha }_1{\alpha }_2}& =0,\hspace{1em}{\alpha }_3=N.\hfill \end{array}$$

Solving the above system of non-linear equations, we obtain the R matrix.

Lemma 3. 

The continuous time Markov chain $X\left(t\right)$ satisfies the stability condition in the Lemma 2; then, the stationary probability vector $\varphi =({\varphi }^{\left(0\right)},{\varphi }^{\left(1\right)},{\varphi }^{\left(2\right)},\dots )$ to the infinitesimal generator matrix P is defined by

$${\varphi }^{\left({\alpha }_1\right)}={\varphi }^{\left(0\right)}{R}^{\left({\alpha }_1\right)},\forall {\alpha }_1=0,1,2,\dots$$

(13)

where R is the minimal non-negative solution of the matrix quadratic Equation (12). The initial stationary probability vector ${\varphi }^{\left(0\right)}$ is obtained by solving

$$\begin{array}{c}\hfill {\varphi }^{\left(0\right)}(A_0+R{A}_2)={\displaystyle \mathit{0}}\end{array}$$

(14)

and

$$\begin{array}{c}\hfill {\varphi }^{\left(0\right)}{(I-R)}^{-1}\mathit{e}=1\end{array}$$

(15)

Proof. 

Since the four-dimensional stochastic process $\left\{X\left(t\right),t\ge 0\right\}$ satisfies $\mathsf{\Phi }P=\mathbf{0}$ and gives

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}{\varphi }^{\left({\alpha }_1\right)}{A}_0+{\varphi }^{({\alpha }_1+1)}{A}_2& =& \mathbf{0},{\alpha }_1=0\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\varphi }^{({\alpha }_1-1)}C+{\varphi }^{\left({\alpha }_1\right)}{A}_1+{\varphi }^{({\alpha }_1+1)}{A}_2& =& \mathbf{0},{\alpha }_1=1,2,\dots \hfill \end{array}$$

(17)

Also the system is stable under the condition in the Lemma 2, and applying Neuts Matrix geometric approach, let us assume that the Equation (13) holds. Now using Equation (13) in (16) and $\varphi \mathbf{e}=1$, we obtain Equations (14) and (15), respectively. Therefore, solving the system of Equations (14) and (15) simultaneously, we obtain ${\varphi }^{\left(0\right)}$. Now we explicitly write Equation (16), which yields:

$$\begin{array}{c}\hfill {\varphi }^{(0,0,0)}[D_{00}^{\left(0\right)}+Z^{1,1}{B}_0]+{\varphi }^{(0,1,1)}{F}_4=\mathbf{0}\end{array}$$

(18)

$$\begin{array}{c}\hfill {\varphi }^{(0,0,0)}{F}_1+{\varphi }^{(0,Q,0)}[D_{0Q}^{\left(0\right)}+Z^{2,2}{B}_0]=\mathbf{0}\end{array}$$

(19)

$$\begin{array}{c}\hfill {\varphi }^{(0,0,0)}{F}_2+{\varphi }^{(0,0,1)}[D_{00}^{\left(1\right)}+Z^{3,3}{B}_1]+{\varphi }^{(0,1,1)}{F}_5=\mathbf{0}\end{array}$$

(20)

$$\begin{array}{c}\hfill {\varphi }^{(0,{\alpha }_1+1,1)}{F}_6+{\varphi }^{(0,{\alpha }_1,1)}[D_{0{\alpha }_1}^{\left(1\right)}+Z^{{\alpha }_1+3,{\alpha }_1+3}{B}_2]=\mathbf{0}{\alpha }_1=1,2,\dots ,Q-1.\end{array}$$

(21)

$$\begin{array}{c}\hfill {\varphi }^{(0,Q,0)}{F}_3+{\varphi }^{(0,0,1)}{F}_7+{\varphi }^{(0,Q,1)}[D_{0Q}^{\left(1\right)}+Z^{Q+3,Q+3}{B}_2]+{\varphi }^{(0,Q+1,1)}{F}_6=\mathbf{0}\end{array}$$

(22)

$$\begin{array}{c}\hfill {\varphi }^{(0,{\alpha }_1-Q,1)}{F}_8+{\varphi }^{(0,{\alpha }_1,1)}[D_{0{\alpha }_1}^{\left(1\right)}+Z^{{\alpha }_1+3,{\alpha }_1+3}{B}_2]+{\varphi }^{(0,{\alpha }_1+1,1)}{F}_6=\mathbf{0}{\alpha }_1=Q+1,Q+2,\dots ,S-1.\end{array}$$

(23)

$$\begin{array}{c}\hfill {\varphi }^{(0,s,1)}{F}_8+{\varphi }^{(0,S,1)}[D_{0S}^{\left(1\right)}+Z^{S+3,S+3}{B}_2]=\mathbf{0}.\end{array}$$

(24)

Solving Equations (18)–(24) and using the normalising condition, we can obtain ${\varphi }^{\left(0\right)}$. □

4. System Performance Measures

In this section, we shall compute the mean number of items, primary customers, and retrial customers; the expected reorder rate; the loss rate of primary customers; the probability that the server is on vacation, busy, or idle; and the expected total cost of the system.

1.

Mean number of Iitems in the Inventory

$$M_I=\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_3=0}^{N}Q{\varphi }^{({\alpha }_1,Q,0,{\alpha }_3)}+\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_2=1}^S\sum _{{\alpha }_3=0}^N{\alpha }_2{\varphi }^{({\alpha }_1,{\alpha }_2,1,{\alpha }_3)}$$

2.

Mean reorder rate

$$M_R=\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_3=1}^N\mu {\varphi }^{({\alpha }_1,s+1,1,{\alpha }_3)}$$

3.

Probability that the server is on vacation

$$p_v=\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_2=0,Q}^{}\sum _{{\alpha }_3=0}^N{\varphi }^{({\alpha }_1,{\alpha }_2,0,{\alpha }_3)}$$

4.

Probability that the server is busy

$$p_b=\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_2=1}^S\sum _{{\alpha }_3=1}^N{\varphi }^{({\alpha }_1,{\alpha }_2,0,{\alpha }_3)}$$

5.

Probability that the server is idle

$$p_i=1-(p_v+p_b)$$

6.

Overall rate of retrial

$$R_{OR}={\theta }_1\sum _{{\alpha }_1=1}^{\infty }\sum _{{\alpha }_2=0,Q}\sum _{{\alpha }_3=0}^{N-1}{\varphi }^{({\alpha }_1,{\alpha }_2,0,{\alpha }_3)}+{\theta }_2\sum _{{\alpha }_1=1}^{\infty }\sum _{{\alpha }_2=0}^S\sum _{{\alpha }_3=0}^{N-1}{\varphi }^{({\alpha }_1,{\alpha }_2,1,{\alpha }_3)}$$

7.

Successful rate of retrial

$$R_{SR}={\theta }_2\sum _{{\alpha }_1=1}^{\infty }\sum _{{\alpha }_2=1}^S\sum _{{\alpha }_3=0}^{N-1}{\varphi }^{({\alpha }_1,{\alpha }_2,1,{\alpha }_3)}$$

8.

Fraction of successful retrial rate

$$F_{SR}=\frac{R_{SR}}{R_{OR}}$$

9.

Mean number of customers in the waiting hall

$$M_{WH}=\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_3=1}^N{\alpha }_3\left[{\varphi }^{({\alpha }_1,0,0,{\alpha }_3)}+{\varphi }^{({\alpha }_1,Q,0,{\alpha }_3)}\right]+\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_2=0}^S\sum _{{\alpha }_3=1}^N{\alpha }_3{\varphi }^{({\alpha }_1,{\alpha }_2,1,{\alpha }_3)}$$

10.

Mean number of customers arriving into the waiting hall

$$M{A}_{WH}={\lambda }_1\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_2=0,Q}\sum _{{\alpha }_3=0}^{N-1}{\varphi }^{({\alpha }_1,{\alpha }_2,0,{\alpha }_3)}+{\lambda }_2\sum _{{\alpha }_1=0}^{\infty }\left(\sum _{{\alpha }_3=0}^{N-1}{\varphi }^{({\alpha }_1,0,1,{\alpha }_2)}+\sum _{{\alpha }_2=1}^S\sum _{{\alpha }_3=0}^{N-1}{\varphi }^{({\alpha }_1,{\alpha }_2,1,{\alpha }_3)}\right)$$

11.

Average waiting time for the customer in the waiting hall

$$WT=\frac{M_{WH}}{M{A}_{WH}}$$

12.

Mean number of customers in the orbit

$$M_O=\sum _{{\alpha }_1=1}^{\infty }\sum _{{\alpha }_3=0}^N{\alpha }_1\left[{\varphi }^{({\alpha }_1,0,0,{\alpha }_3)}+{\varphi }^{({\alpha }_1,Q,0,{\alpha }_3)}\right]+\sum _{{\alpha }_1=1}^{\infty }\sum _{{\alpha }_2=1}^S{\alpha }_1{\varphi }^{({\alpha }_1,{\alpha }_2,1,0)}+\sum _{{\alpha }_1=1}^{\infty }\sum _{{\alpha }_2=0}^S\sum _{{\alpha }_3=1}^N{\alpha }_1{\varphi }^{({\alpha }_1,{\alpha }_2,1,{\alpha }_3)}$$

13.

Mean number of customers arriving into the orbit

$$M{A}_O={\theta }_1\sum _{{\alpha }_1=0}^{\infty }\sum _{{\alpha }_2=0,Q}{\varphi }^{({\alpha }_1,{\alpha }_2,0,N)}+{\theta }_2\sum _{{\alpha }_1=0}^{\infty }\left({\varphi }^{({\alpha }_1,0,1,N)}+\sum _{{\alpha }_2=1}^S{\varphi }^{({\alpha }_1,{\alpha }_2,1,N)}\right)$$

14.

Waiting time for the customer in the orbit

$$WTO=\frac{M_O}{M{A}_O}$$

Total Cost Analysis

The expected total cost of the considered model is defined by

$$TC=c_h{M}_I+c_s{M}_R+c_{wo}{M}_O+c_w{M}_{WH}$$

where

$c_h$—Holding cost per unit item;

$c_s$—Set-up cost per order;

$c_{wo}$—Waiting cost for the customer in the orbit;

$c_w$—Waiting cost for the customer in the waiting hall.

5. Numerical Examples

In this section, we have shown the importance of the assumed model using a numerical illustration. Though we have explained the concept mathematically, its applications are seen only with the help of numerical results. Based on the need for numerical examples in business, necessary results are obtained to support the model.

For the purpose of research, we have fixed parameter values $S=32$, $s=10$, $Q=22$, $N=5$, $mu=10$, $\beta =4$, $\gamma =2$, ${\theta }_1=0.6$, ${\theta }_2=2.5$, ${\lambda }_1=2.5$, ${\lambda }_2=3.5$, $ch=0.01$, $cs=0.5$, $cwo=1$, and $cw=1.3$, and obtained the results.

Example 1. 

This example explains the effects of parameters ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N on the total cost, which are provided in the Table 1.

1.

We observe that the total cost increases whenever ${\lambda }_1$ and ${\lambda }_2$ increase. Since the arrival rate is increasing, there will be a higher number of customers arriving in the waiting hall. Thus, the total cost will be increased along with the waiting cost.

2.

The retrial rates ${\theta }_1$ and ${\theta }_2$ impact the total cost by reducing it as they increase. This is because, when the retrial rate is increased, customers leave the orbit. Thus the product value of $M_O$ and the waiting cost for them is decreased, and this will yield the decreases of the total cost.

3.

It is clear that the total cost decreases as the service rate μ increases. As μ increases, the average service time per customer is decreased, and the waiting hall size is also decreased simultaneously. Therefore, the waiting cost for the $M_{WH}$ at the time is decreased, which causes the decrease in the total cost.

4.

Observing the result, we come to know that if the vacation completion rate increases, the respective total decreases. This is because the duration of the server being on vacation decreases as γ increases, and the server will be available to provide service to the customers. In this situation, the number of customers in the waiting hall will decrease. Then the waiting cost for the $M_{WH}$ at the time will decrease that yields reduce of total cost.

5.

It is known that total cost increases as N increases. This is because, when the waiting hall size is increased, $M_{WH}$ will be increased. So the product value of the $M_{WH}$ and the waiting cost for them increases the total cost.

6.

Whenever the reorder rate β increases, the total cost decreases. This means the average replenishment time is decreased when β increases. Therefore, the setup cost per order is decreased, which causes a decrease in the total cost.

Example 2. 

In this example, we have discussed the average waiting time of a customer in the waiting hall and the orbit (Table 2 and Table 3).

1.

From Table 2, the average waiting time of customers in the waiting hall and orbit increases as the arrival rates ${\lambda }_1$ and ${\lambda }_2$ increase, because more number of customers will be arriving at the waiting hall due to the increment in the arrival rates. Similarly, when ${\lambda }_1$ and ${\lambda }_2$ increase, the customer in the orbit increases. This results in an increase in waiting time for the primary and orbital customers.

2.

It is clear that the $M_{WH}$ increases and the number of customers in the the orbit decreases if the retrial rates ${\theta }_1$ and ${\theta }_2$ increase. This causes an increase in the average waiting time for the customer in the waiting hall and a decrease in the average waiting time for the customer in the orbit.

3.

While increasing the service rate μ, the average service time per customer is increased in Table 3. Consequently, the number of customers in the waiting hall as well as in the orbit has decreased. So the average waiting time of both customers has decreased.

4.

We observe that the average waiting time of the primary and retrial customers reduces when the reorder rate is increased. In this case, the mean reorder time is decreased. That results in a decrease in the $M_{WH}$. Similarly, the number of customers in orbit also decreases. This explains the decrease in the average waiting time of both customers as β increases.

5.

One part of the average waiting time of a customer always depends upon the waiting hall size. Increasing the waiting hall size increases $M_{WH}$, which increases the waiting time of customers in the waiting hall. For the retrial customers, as there are more place in the waiting hall, $M_O$ decreases. So the waiting time of customers in the orbit decreases while increasing N.

Table 1. Total cost under the parameter variation.

${\mathit{\theta }}_2$			2	2.5	3	$\mathit{\mu }$			9	10.5	12
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$				$\mathbf{\gamma }$	$N$	$\mathbf{\beta }$
0.4		4	1.16049626	1.15994888	1.15967796			2.5	1.566933	1.274744	1.088836
	3.3	5	1.60073258	1.59847405	1.59737999		4	4	1.564398	1.272200	1.086288
		6	2.20734817	2.20021918	2.19679518			5.5	1.563741	1.271203	1.085071
		4	1.16051381	1.15996627	1.15969530			2.5	1.578102	1.275889	1.087037
	3.5	5	1.60079059	1.59853154	1.59743728	1.8	5	4	1.575629	1.273390	1.084524
		6	2.20748564	2.20035502	2.19693028			5.5	1.575577	1.272823	1.083637
		4	1.16053124	1.15998354	1.15971251			2.5	1.589663	1.279158	1.087767
	3.7	5	1.60084824	1.59858862	1.59749416		6	4	1.587715	1.276695	1.085279
		6	2.20762226	2.20048991	2.19706439			5.5	1.587242	1.276489	1.084668
0.6		4	1.16049626	1.15994887	1.15967794			2.5	1.566922	1.274731	1.088822
	3.3	5	1.60073247	1.59847384	1.59737974		4	4	1.564333	1.272129	1.086212
		6	2.20734703	2.20021770	2.19679362			5.5	1.563282	1.270707	1.084546
		4	1.16051380	1.15996626	1.15969528			2.5	1.578088	1.275874	1.087021
	3.5	5	1.60079049	1.59853134	1.59743705	2.05	5	4	1.575555	1.273311	1.084440
		6	2.20748455	2.20035359	2.19692878			5.5	1.575071	1.272280	1.083068
		4	1.16053123	1.15998353	1.15971249			2.5	1.589648	1.279142	1.087749
	3.7	5	1.60084814	1.59858843	1.59749393		6	4	1.587167	1.276609	1.085189
		6	2.20762122	2.20048854	2.19706295			5.5	1.587160	1.275906	1.084059
0.8		4	1.16049625	1.15994885	1.15967791			2.5	1.566913	1.274721	1.088812
	3.3	5	1.60073235	1.59847360	1.59737946		4	4	1.564285	1.272077	1.086156
		6	2.20734581	2.20021609	2.19679192			5.5	1.562947	1.270345	1.084165
		4	1.16051379	1.15996625	1.15969526			2.5	1.578078	1.275863	1.087010
	3.5	5	1.60079037	1.59853111	1.59743678	2.3	5	4	1.575502	1.273253	1.084379
		6	2.20748338	2.20035205	2.19692715			5.5	1.574705	1.271889	1.082657
		4	1.16053122	1.15998352	1.15971247			2.5	1.589636	1.279129	1.087736
	3.7	5	1.60084803	1.59858821	1.59749368		6	4	1.587101	1.276546	1.085124
		6	2.20762009	2.20048706	2.19706139			5.5	1.586773	1.275488	1.083623

Table 1. Total cost under the parameter variation.

${\mathit{\theta }}_2$			2	2.5	3	$\mathit{\mu }$			9	10.5	12
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$				$\mathbf{\gamma }$	$N$	$\mathbf{\beta }$
0.4		4	1.16049626	1.15994888	1.15967796			2.5	1.566933	1.274744	1.088836
	3.3	5	1.60073258	1.59847405	1.59737999		4	4	1.564398	1.272200	1.086288
		6	2.20734817	2.20021918	2.19679518			5.5	1.563741	1.271203	1.085071
		4	1.16051381	1.15996627	1.15969530			2.5	1.578102	1.275889	1.087037
	3.5	5	1.60079059	1.59853154	1.59743728	1.8	5	4	1.575629	1.273390	1.084524
		6	2.20748564	2.20035502	2.19693028			5.5	1.575577	1.272823	1.083637
		4	1.16053124	1.15998354	1.15971251			2.5	1.589663	1.279158	1.087767
	3.7	5	1.60084824	1.59858862	1.59749416		6	4	1.587715	1.276695	1.085279
		6	2.20762226	2.20048991	2.19706439			5.5	1.587242	1.276489	1.084668
0.6		4	1.16049626	1.15994887	1.15967794			2.5	1.566922	1.274731	1.088822
	3.3	5	1.60073247	1.59847384	1.59737974		4	4	1.564333	1.272129	1.086212
		6	2.20734703	2.20021770	2.19679362			5.5	1.563282	1.270707	1.084546
		4	1.16051380	1.15996626	1.15969528			2.5	1.578088	1.275874	1.087021
	3.5	5	1.60079049	1.59853134	1.59743705	2.05	5	4	1.575555	1.273311	1.084440
		6	2.20748455	2.20035359	2.19692878			5.5	1.575071	1.272280	1.083068
		4	1.16053123	1.15998353	1.15971249			2.5	1.589648	1.279142	1.087749
	3.7	5	1.60084814	1.59858843	1.59749393		6	4	1.587167	1.276609	1.085189
		6	2.20762122	2.20048854	2.19706295			5.5	1.587160	1.275906	1.084059
0.8		4	1.16049625	1.15994885	1.15967791			2.5	1.566913	1.274721	1.088812
	3.3	5	1.60073235	1.59847360	1.59737946		4	4	1.564285	1.272077	1.086156
		6	2.20734581	2.20021609	2.19679192			5.5	1.562947	1.270345	1.084165
		4	1.16051379	1.15996625	1.15969526			2.5	1.578078	1.275863	1.087010
	3.5	5	1.60079037	1.59853111	1.59743678	2.3	5	4	1.575502	1.273253	1.084379
		6	2.20748338	2.20035205	2.19692715			5.5	1.574705	1.271889	1.082657
		4	1.16053122	1.15998352	1.15971247			2.5	1.589636	1.279129	1.087736
	3.7	5	1.60084803	1.59858821	1.59749368		6	4	1.587101	1.276546	1.085124
		6	2.20762009	2.20048706	2.19706139			5.5	1.586773	1.275488	1.083623

Table 2. WT, WTO under the variation of parameters ${\lambda }_1,{\lambda }_2$, ${\theta }_1,{\theta }_2$.

${\mathit{\lambda }}_2$			4		5		6
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\theta }}_{\mathbf{2}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	WT	WTO	WT	WTO	WT	WTO
		3.3	0.1640018	0.601369	0.1914943	0.603069	0.2267440	0.605800
	2	3.5	0.1640039	0.601379	0.1914997	0.603077	0.2267548	0.605805
		3.7	0.1640060	0.601388	0.1915051	0.603085	0.2267655	0.605811
		3.3	0.1641697	0.501025	0.1919572	0.502266	0.2278050	0.504274
0.4	2.5	3.5	0.1641718	0.501036	0.1919628	0.502275	0.2278162	0.504280
		3.7	0.1641739	0.501045	0.1919683	0.502283	0.2278271	0.504285
		3.3	0.1643207	0.434139	0.1923805	0.435084	0.2287917	0.436624
	3	3.5	0.1643228	0.434149	0.1923862	0.435093	0.2288031	0.436630
		3.7	0.1643249	0.434159	0.1923918	0.435101	0.2288143	0.436636
		3.3	0.1640019	0.601351	0.1914947	0.603039	0.2267454	0.605758
	2	3.5	0.1640040	0.601361	0.1915001	0.603048	0.2267562	0.605764
		3.7	0.1640060	0.601371	0.1915055	0.603056	0.2267668	0.605770
		3.3	0.1641697	0.501012	0.1919575	0.502244	0.2278062	0.504243
0.6	2.5	3.5	0.1641719	0.501023	0.1919631	0.502253	0.2278173	0.504250
		3.7	0.1641739	0.501033	0.1919686	0.502262	0.2278282	0.504256
		3.3	0.1643207	0.434129	0.1923808	0.435068	0.2287927	0.436601
	3	3.5	0.1643229	0.434139	0.1923865	0.435077	0.2288041	0.436607
		3.7	0.1643250	0.434150	0.1923921	0.435085	0.2288152	0.436614
		3.3	0.1640020	0.601335	0.1914950	0.603012	0.2267466	0.605719
	2	3.5	0.1640041	0.601346	0.1915005	0.603021	0.2267574	0.605726
		3.7	0.1640061	0.601356	0.1915058	0.603030	0.2267680	0.605732
		3.3	0.1641698	0.501000	0.1919578	0.502224	0.2278073	0.504215
0.8	2.5	3.5	0.1641719	0.501011	0.1919634	0.502234	0.2278183	0.504222
		3.7	0.1641740	0.501021	0.1919689	0.502243	0.2278292	0.504229
		3.3	0.1643208	0.434120	0.1923810	0.435052	0.2287937	0.436579
	3	3.5	0.1643229	0.434130	0.1923867	0.435062	0.2288050	0.436586
		3.7	0.1643251	0.434141	0.1923923	0.435071	0.2288161	0.436593

Table 2. WT, WTO under the variation of parameters ${\lambda }_1,{\lambda }_2$, ${\theta }_1,{\theta }_2$.

${\mathit{\lambda }}_2$			4		5		6
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\theta }}_{\mathbf{2}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	WT	WTO	WT	WTO	WT	WTO
		3.3	0.1640018	0.601369	0.1914943	0.603069	0.2267440	0.605800
	2	3.5	0.1640039	0.601379	0.1914997	0.603077	0.2267548	0.605805
		3.7	0.1640060	0.601388	0.1915051	0.603085	0.2267655	0.605811
		3.3	0.1641697	0.501025	0.1919572	0.502266	0.2278050	0.504274
0.4	2.5	3.5	0.1641718	0.501036	0.1919628	0.502275	0.2278162	0.504280
		3.7	0.1641739	0.501045	0.1919683	0.502283	0.2278271	0.504285
		3.3	0.1643207	0.434139	0.1923805	0.435084	0.2287917	0.436624
	3	3.5	0.1643228	0.434149	0.1923862	0.435093	0.2288031	0.436630
		3.7	0.1643249	0.434159	0.1923918	0.435101	0.2288143	0.436636
		3.3	0.1640019	0.601351	0.1914947	0.603039	0.2267454	0.605758
	2	3.5	0.1640040	0.601361	0.1915001	0.603048	0.2267562	0.605764
		3.7	0.1640060	0.601371	0.1915055	0.603056	0.2267668	0.605770
		3.3	0.1641697	0.501012	0.1919575	0.502244	0.2278062	0.504243
0.6	2.5	3.5	0.1641719	0.501023	0.1919631	0.502253	0.2278173	0.504250
		3.7	0.1641739	0.501033	0.1919686	0.502262	0.2278282	0.504256
		3.3	0.1643207	0.434129	0.1923808	0.435068	0.2287927	0.436601
	3	3.5	0.1643229	0.434139	0.1923865	0.435077	0.2288041	0.436607
		3.7	0.1643250	0.434150	0.1923921	0.435085	0.2288152	0.436614
		3.3	0.1640020	0.601335	0.1914950	0.603012	0.2267466	0.605719
	2	3.5	0.1640041	0.601346	0.1915005	0.603021	0.2267574	0.605726
		3.7	0.1640061	0.601356	0.1915058	0.603030	0.2267680	0.605732
		3.3	0.1641698	0.501000	0.1919578	0.502224	0.2278073	0.504215
0.8	2.5	3.5	0.1641719	0.501011	0.1919634	0.502234	0.2278183	0.504222
		3.7	0.1641740	0.501021	0.1919689	0.502243	0.2278292	0.504229
		3.3	0.1643208	0.434120	0.1923810	0.435052	0.2287937	0.436579
	3	3.5	0.1643229	0.434130	0.1923867	0.435062	0.2288050	0.436586
		3.7	0.1643251	0.434141	0.1923923	0.435071	0.2288161	0.436593

Table 3. WT, WTO under the parameter variation ${\lambda }_1,{\lambda }_2$, ${\theta }_1,{\theta }_2$.

$\mathit{\beta }$			2.5		4		5.5
$N$	$\mathbf{\gamma }$	$\mathbf{\mu }$	WT	WTO	WT	WTO	WT	WTO
4		9	0.208426	0.517491	0.207456	0.516128	0.207370	0.515991
	1.8	10.5	0.160917	0.500893	0.160030	0.498722	0.159947	0.498502
		12	0.130799	0.489298	0.129960	0.485964	0.129879	0.485622
		9	0.208354	0.517346	0.207446	0.516107	0.207368	0.515987
	2.05	10.5	0.160842	0.500620	0.160019	0.498682	0.159945	0.498494
		12	0.130721	0.488828	0.129949	0.485893	0.129877	0.485609
		9	0.208207	0.517077	0.207425	0.516067	0.207364	0.515980
	2.3	10.5	0.160691	0.500134	0.159997	0.498608	0.159941	0.498480
		12	0.130566	0.488010	0.129926	0.485767	0.129873	0.485585
5		9	0.214582	0.514856	0.213481	0.513138	0.213381	0.512962
	1.8	10.5	0.164254	0.499808	0.163257	0.496706	0.163164	0.496390
		12	0.132765	0.490032	0.131829	0.484682	0.131739	0.484134
		9	0.214499	0.514650	0.213469	0.513107	0.213379	0.512956
	2.05	10.5	0.164168	0.499361	0.163245	0.496637	0.163161	0.496376
		12	0.132677	0.489167	0.131816	0.484545	0.131736	0.484106
		9	0.214439	0.514510	0.213460	0.513085	0.213378	0.512952
	2.3	10.5	0.164106	0.499064	0.163236	0.496590	0.163160	0.496367
		12	0.132612	0.488597	0.131807	0.484453	0.131735	0.484088
6		9	0.218524	0.514227	0.217311	0.512080	0.217201	0.511860
	1.8	10.5	0.166129	0.500579	0.165043	0.496164	0.164941	0.495718
		12	0.133756	0.493127	0.132744	0.484633	0.132646	0.483756
		9	0.218432	0.513941	0.217298	0.512034	0.217198	0.511850
	2.05	10.5	0.166034	0.499867	0.165029	0.496049	0.164938	0.495695
		12	0.133658	0.491602	0.132729	0.484373	0.132643	0.483704
		9	0.218366	0.513755	0.217288	0.512003	0.217196	0.511844
	2.3	10.5	0.165965	0.499413	0.165019	0.495975	0.164936	0.495680
		12	0.133587	0.490635	0.132719	0.484207	0.132641	0.483670

Table 3. WT, WTO under the parameter variation ${\lambda }_1,{\lambda }_2$, ${\theta }_1,{\theta }_2$.

$\mathit{\beta }$			2.5		4		5.5
$N$	$\mathbf{\gamma }$	$\mathbf{\mu }$	WT	WTO	WT	WTO	WT	WTO
4		9	0.208426	0.517491	0.207456	0.516128	0.207370	0.515991
	1.8	10.5	0.160917	0.500893	0.160030	0.498722	0.159947	0.498502
		12	0.130799	0.489298	0.129960	0.485964	0.129879	0.485622
		9	0.208354	0.517346	0.207446	0.516107	0.207368	0.515987
	2.05	10.5	0.160842	0.500620	0.160019	0.498682	0.159945	0.498494
		12	0.130721	0.488828	0.129949	0.485893	0.129877	0.485609
		9	0.208207	0.517077	0.207425	0.516067	0.207364	0.515980
	2.3	10.5	0.160691	0.500134	0.159997	0.498608	0.159941	0.498480
		12	0.130566	0.488010	0.129926	0.485767	0.129873	0.485585
5		9	0.214582	0.514856	0.213481	0.513138	0.213381	0.512962
	1.8	10.5	0.164254	0.499808	0.163257	0.496706	0.163164	0.496390
		12	0.132765	0.490032	0.131829	0.484682	0.131739	0.484134
		9	0.214499	0.514650	0.213469	0.513107	0.213379	0.512956
	2.05	10.5	0.164168	0.499361	0.163245	0.496637	0.163161	0.496376
		12	0.132677	0.489167	0.131816	0.484545	0.131736	0.484106
		9	0.214439	0.514510	0.213460	0.513085	0.213378	0.512952
	2.3	10.5	0.164106	0.499064	0.163236	0.496590	0.163160	0.496367
		12	0.132612	0.488597	0.131807	0.484453	0.131735	0.484088
6		9	0.218524	0.514227	0.217311	0.512080	0.217201	0.511860
	1.8	10.5	0.166129	0.500579	0.165043	0.496164	0.164941	0.495718
		12	0.133756	0.493127	0.132744	0.484633	0.132646	0.483756
		9	0.218432	0.513941	0.217298	0.512034	0.217198	0.511850
	2.05	10.5	0.166034	0.499867	0.165029	0.496049	0.164938	0.495695
		12	0.133658	0.491602	0.132729	0.484373	0.132643	0.483704
		9	0.218366	0.513755	0.217288	0.512003	0.217196	0.511844
	2.3	10.5	0.165965	0.499413	0.165019	0.495975	0.164936	0.495680
		12	0.133587	0.490635	0.132719	0.484207	0.132641	0.483670

Example 3. 

In this example, we explained the fraction of time server being on vacation by varying the parameters ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N in Table 4.

1.

While the server is on vacation, the fraction of time the server is on vacation decreases as the arrival rate increases. While increasing ${\lambda }_1$, the number of customers in the waiting hall increases. So the customer’s arrival during the server’s vacation leads the server to work. Thus, the possibility of the server being on vacation decreases.

2.

While increasing the arrival rate of customers during the server’s normal mode, the fraction of time the server being on vacation increases. As ${\lambda }_2$ increases, the number of primary customers increases in the waiting hall, which leads to stock-out frequently and the server going on vacation frequently.

3.

The fraction of time the server is on vacation decreases if the arrival rate of retrial customers during the server’s vacation increases. Because the $M_{WH}$ increases due to the increase of ${\theta }_1$. So the customer’s arrival in the waiting hall during vacation leads the server to work Â.

4.

 As ${\theta }_2$ increases, the number of customers increases in the waiting hall, which leads to stock-out often, and the server goes on vacation frequently, such that the fraction of time the server is on vacation increases.

5.

Since the service rate increases, the fraction of time that the server is on vacation increases. Due to the increment in the rate μ, $M_O$ and items in the inventory decrease. So, the server can take a vacation frequently.

6.

Whenever the replenishment rate increases, the fraction of time a server is on vacation decreases. If we increase the value of β, the replenishment time is decreased. So server has to work for more time, which decreases the fraction of time the server is on vacation.

7.

By increasing the vacation completion rate, the fraction of time a server is on vacation decreases. Since the rate γ increases, the duration of server on vacation is decreased. So the server has to return to the normal mode often, which confirms the decrease in the fraction of time the server is on vacation.

8.

The number of customers in the waiting hall increases upon increasing the waiting hall size, which leads to a stock out often as more customers can be accommodated in the waiting hall. So the possibility of a server taking a vacation is as high as inventory.

If the server is on vacation for a long time, a business cannot be more profitable. So, providing service at a fast rate, completing the vacation time sooner, and having a fast reorder will help the business run without loss.

Table 4. $p_v$ under variation of ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N.

${\mathit{\theta }}_2$			2	2.5	3	$\mathit{\mu }$			9	10.5	12
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$				$\mathbf{\gamma }$	$N$	$\mathbf{\beta }$
		4	0.0000671	0.0000673	0.0000675			2.5	0.0008935	0.0010153	0.0011096
	3.3	5	0.0002016	0.0002022	0.0002027		4	4	0.0001142	0.0001320	0.0001461
		6	0.0004277	0.0004285	0.0004289			5.5	0.0000196	0.0000232	0.0000262
		4	0.0000667	0.0000669	0.0000671			2.5	0.0008958	0.0010187	0.0011130
0.4	3.5	5	0.0002004	0.0002011	0.0002016	1.8	5	4	0.0001187	0.0001361	0.0001497
		6	0.0004253	0.0004261	0.0004265			5.5	0.0000214	0.0000249	0.0000277
		4	0.0000663	0.0000665	0.0000667			2.5	0.0008965	0.0010202	0.0011144
	3.7	5	0.0001993	0.0002000	0.0002005		6	4	0.0001212	0.0001383	0.0001515
		6	0.0004230	0.0004238	0.0004242			5.5	0.0000225	0.0000259	0.0000285
		4	0.00006704	0.0000673	0.0000674			2.5	0.0008093	0.0009198	0.0010053
	3.3	5	0.00020147	0.0002022	0.0002027		4	4	0.0001029	0.0001190	0.0001317
		6	0.00042734	0.0004282	0.0004286			5.5	0.0000176	0.0000209	0.0000235
		4	0.00006666	0.0000669	0.0000671			2.5	0.0008115	0.0009230	0.0010084
0.6	3.5	5	0.00020033	0.0002010	0.0002015	2.05	5	4	0.0001070	0.0001227	0.0001350
		6	0.00042496	0.0004258	0.0004262			5.5	0.0000192	0.0000224	0.0000249
		4	0.00006630	0.0000665	0.0000667			2.5	0.0008123	0.0009244	0.0010097
	3.7	5	0.00019925	0.0001999	0.0002004		6	4	0.0001092	0.0001247	0.0001366
		6	0.00042270	0.0004235	0.0004239			5.5	0.0000202	0.0000232	0.0000256
		4	0.0000670	0.0000673	0.0000674			2.5	0.0007435	0.0008450	0.0009236
	3.3	5	0.0002014	0.0002021	0.0002026		4	4	0.0000941	0.0001088	0.0001204
		6	0.0004270	0.0004279	0.0004284			5.5	0.0000161	0.0000190	0.0000214
		4	0.0000666	0.0000669	0.0000670			2.5	0.0007456	0.0008481	0.0009266
0.8	3.5	5	0.0002003	0.0002010	0.0002015	2.3	5	4	0.0000978	0.0001122	0.0001234
		6	0.0004246	0.0004255	0.0004260			5.5	0.0000175	0.0000204	0.0000227
		4	0.0000663	0.0000665	0.0000667			2.5	0.0007463	0.0008494	0.0009278
	3.7	5	0.0001992	0.0001999	0.0002004		6	4	0.0000999	0.0001141	0.0001249
		6	0.0004224	0.0004233	0.0004237			5.5	0.0000184	0.0000212	0.0000233

Table 4. $p_v$ under variation of ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N.

${\mathit{\theta }}_2$			2	2.5	3	$\mathit{\mu }$			9	10.5	12
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$				$\mathbf{\gamma }$	$N$	$\mathbf{\beta }$
		4	0.0000671	0.0000673	0.0000675			2.5	0.0008935	0.0010153	0.0011096
	3.3	5	0.0002016	0.0002022	0.0002027		4	4	0.0001142	0.0001320	0.0001461
		6	0.0004277	0.0004285	0.0004289			5.5	0.0000196	0.0000232	0.0000262
		4	0.0000667	0.0000669	0.0000671			2.5	0.0008958	0.0010187	0.0011130
0.4	3.5	5	0.0002004	0.0002011	0.0002016	1.8	5	4	0.0001187	0.0001361	0.0001497
		6	0.0004253	0.0004261	0.0004265			5.5	0.0000214	0.0000249	0.0000277
		4	0.0000663	0.0000665	0.0000667			2.5	0.0008965	0.0010202	0.0011144
	3.7	5	0.0001993	0.0002000	0.0002005		6	4	0.0001212	0.0001383	0.0001515
		6	0.0004230	0.0004238	0.0004242			5.5	0.0000225	0.0000259	0.0000285
		4	0.00006704	0.0000673	0.0000674			2.5	0.0008093	0.0009198	0.0010053
	3.3	5	0.00020147	0.0002022	0.0002027		4	4	0.0001029	0.0001190	0.0001317
		6	0.00042734	0.0004282	0.0004286			5.5	0.0000176	0.0000209	0.0000235
		4	0.00006666	0.0000669	0.0000671			2.5	0.0008115	0.0009230	0.0010084
0.6	3.5	5	0.00020033	0.0002010	0.0002015	2.05	5	4	0.0001070	0.0001227	0.0001350
		6	0.00042496	0.0004258	0.0004262			5.5	0.0000192	0.0000224	0.0000249
		4	0.00006630	0.0000665	0.0000667			2.5	0.0008123	0.0009244	0.0010097
	3.7	5	0.00019925	0.0001999	0.0002004		6	4	0.0001092	0.0001247	0.0001366
		6	0.00042270	0.0004235	0.0004239			5.5	0.0000202	0.0000232	0.0000256
		4	0.0000670	0.0000673	0.0000674			2.5	0.0007435	0.0008450	0.0009236
	3.3	5	0.0002014	0.0002021	0.0002026		4	4	0.0000941	0.0001088	0.0001204
		6	0.0004270	0.0004279	0.0004284			5.5	0.0000161	0.0000190	0.0000214
		4	0.0000666	0.0000669	0.0000670			2.5	0.0007456	0.0008481	0.0009266
0.8	3.5	5	0.0002003	0.0002010	0.0002015	2.3	5	4	0.0000978	0.0001122	0.0001234
		6	0.0004246	0.0004255	0.0004260			5.5	0.0000175	0.0000204	0.0000227
		4	0.0000663	0.0000665	0.0000667			2.5	0.0007463	0.0008494	0.0009278
	3.7	5	0.0001992	0.0001999	0.0002004		6	4	0.0000999	0.0001141	0.0001249
		6	0.0004224	0.0004233	0.0004237			5.5	0.0000184	0.0000212	0.0000233

Example 4. 

In this example, we have presented the result of $F_{SR}$ varying the parameters ${\lambda }_1,{\lambda }_2,$${\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N in Table 5.

1.

The fraction of successful retrial rate decreases as arrival rates ${\lambda }_1$ and ${\lambda }_2$ increase. Due to the increase in the arrival rate, the number of customers in the orbit increases. This decreases the fraction of successful retrials.

2.

We observe that the fraction of successful retrial rate $F_{SR}$ decreases whenever retrial rate increases. While increasing the retrial rate, $OR$ and $SR$ increase. However, the increment is high in the overall retrial rate compared to the successful retrial rate.

3.

While increasing the service rate, the fraction of successful retrials also increases. This is because an increment in μ decreases the $M_O$ as the service time per customer decreases, which increases the $F_{SR}$.

4.

We can note that whenever the reorder rate increases, the fraction of successful retrials also increases. Since β increases, the replenishment time decreases. So the number of customers in the orbit decreases, which increases the $F_{SR}$.

5.

The fraction of successful retrials is increased upon increasing the rate of vacation completion. The reason behind this is the increment in γ decreases the duration of server in the vacation. Then the customer switches to normal mode more frequently.

6.

The result shows us that the fraction of successful retrials increases while we increase the waiting hall size. Since the waiting hall size is increased, $M_O$ is decreased. Hence, the $F_{SR}$ is increased whenever N is increased.

A good businessman always tries to provide satisfactory service to every customer. From the above, one can decide whether to cover the retrial customers or not. In order to provide service to the orbital customer, he has to increase the service rate so that both customers obtain service quickly, and the availability of space in the waiting hall will be possible. Similarly, concerning the rate of reorder and vacation completion, it will help to increase the fraction of successful retrial rate.

Table 5. $F_{SR}$ under the variation of ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N.

${\mathit{\theta }}_2$			2	2.5	3	$\mathit{\mu }$			9	10.5	12
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$				$\mathbf{\gamma }$	$N$	$\mathbf{\beta }$
		4	0.8327968	0.799586	0.7689086			2.5	0.7742734	0.8008768	0.8212734
	3.3	5	0.8317090	0.798689	0.7681589		4	4	0.7751976	0.8024012	0.8236911
		6	0.8296627	0.797000	0.7667535			5.5	0.7752443	0.8024732	0.8238009
		4	0.8327856	0.799572	0.7688920			2.5	0.7785439	0.8035304	0.8221796
0.4	3.5	5	0.8317022	0.798680	0.7681485	1.8	5	4	0.7797848	0.8058368	0.8262659
		6	0.8296593	0.796995	0.7667478			5.5	0.7798407	0.8059306	0.8264244
		4	0.8327764	0.799560	0.7688779			2.5	0.7798401	0.8035217	0.8202201
	3.7	5	0.8316967	0.798672	0.7681398		6	4	0.7814781	0.8069633	0.8269889
		6	0.8296568	0.796991	0.7667433			5.5	0.7815432	0.8070833	0.8272157
		4	0.832756	0.7995490	0.768875			2.5	0.774287	0.8008994	0.821306
	3.3	5	0.831679	0.7986623	0.768135		4	4	0.775198	0.8024020	0.823691
		6	0.829643	0.7969820	0.766738			5.5	0.775244	0.8024733	0.823801
		4	0.832750	0.7995392	0.768863			2.5	0.778567	0.8035765	0.822256
0.6	3.5	5	0.831676	0.7986567	0.768128	2.05	5	4	0.779787	0.8058397	0.826268
		6	0.829642	0.7969794	0.766734			5.5	0.779841	0.8059310	0.826425
		4	0.832745	0.7995316	0.768852			2.5	0.779876	0.8036047	0.820384
	3.7	5	0.831674	0.7986525	0.768122		6	4	0.781481	0.8069694	0.826996
		6	0.829641	0.7969778	0.766731			5.5	0.781544	0.8070842	0.827217
		4	0.832715	0.799512	0.768841			2.5	0.7743052	0.800929	0.8213452
	3.3	5	0.831650	0.798636	0.768111		4	4	0.7751995	0.802403	0.8236914
		6	0.829622	0.796964	0.766722			5.5	0.7752445	0.802473	0.8238005
		4	0.832714	0.799507	0.768833			2.5	0.7786020	0.803642	0.8223587
0.8	3.5	5	0.831650	0.798633	0.768107	2.3	5	4	0.7797897	0.805844	0.8262726
		6	0.829623	0.796964	0.766720			5.5	0.7798414	0.805932	0.8264249
		4	0.832714	0.799503	0.768827			2.5	0.7799325	0.803728	0.8206148
	3.7	5	0.831652	0.798632	0.768104		6	4	0.7814870	0.806980	0.8270071
		6	0.829625	0.796964	0.766719			5.5	0.7815446	0.807086	0.8272181

Table 5. $F_{SR}$ under the variation of ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\mu ,\gamma ,\beta$, and N.

${\mathit{\theta }}_2$			2	2.5	3	$\mathit{\mu }$			9	10.5	12
${\mathbf{\theta }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{1}}$	${\mathbf{\lambda }}_{\mathbf{2}}$				$\mathbf{\gamma }$	$N$	$\mathbf{\beta }$
		4	0.8327968	0.799586	0.7689086			2.5	0.7742734	0.8008768	0.8212734
	3.3	5	0.8317090	0.798689	0.7681589		4	4	0.7751976	0.8024012	0.8236911
		6	0.8296627	0.797000	0.7667535			5.5	0.7752443	0.8024732	0.8238009
		4	0.8327856	0.799572	0.7688920			2.5	0.7785439	0.8035304	0.8221796
0.4	3.5	5	0.8317022	0.798680	0.7681485	1.8	5	4	0.7797848	0.8058368	0.8262659
		6	0.8296593	0.796995	0.7667478			5.5	0.7798407	0.8059306	0.8264244
		4	0.8327764	0.799560	0.7688779			2.5	0.7798401	0.8035217	0.8202201
	3.7	5	0.8316967	0.798672	0.7681398		6	4	0.7814781	0.8069633	0.8269889
		6	0.8296568	0.796991	0.7667433			5.5	0.7815432	0.8070833	0.8272157
		4	0.832756	0.7995490	0.768875			2.5	0.774287	0.8008994	0.821306
	3.3	5	0.831679	0.7986623	0.768135		4	4	0.775198	0.8024020	0.823691
		6	0.829643	0.7969820	0.766738			5.5	0.775244	0.8024733	0.823801
		4	0.832750	0.7995392	0.768863			2.5	0.778567	0.8035765	0.822256
0.6	3.5	5	0.831676	0.7986567	0.768128	2.05	5	4	0.779787	0.8058397	0.826268
		6	0.829642	0.7969794	0.766734			5.5	0.779841	0.8059310	0.826425
		4	0.832745	0.7995316	0.768852			2.5	0.779876	0.8036047	0.820384
	3.7	5	0.831674	0.7986525	0.768122		6	4	0.781481	0.8069694	0.826996
		6	0.829641	0.7969778	0.766731			5.5	0.781544	0.8070842	0.827217
		4	0.832715	0.799512	0.768841			2.5	0.7743052	0.800929	0.8213452
	3.3	5	0.831650	0.798636	0.768111		4	4	0.7751995	0.802403	0.8236914
		6	0.829622	0.796964	0.766722			5.5	0.7752445	0.802473	0.8238005
		4	0.832714	0.799507	0.768833			2.5	0.7786020	0.803642	0.8223587
0.8	3.5	5	0.831650	0.798633	0.768107	2.3	5	4	0.7797897	0.805844	0.8262726
		6	0.829623	0.796964	0.766720			5.5	0.7798414	0.805932	0.8264249
		4	0.832714	0.799503	0.768827			2.5	0.7799325	0.803728	0.8206148
	3.7	5	0.831652	0.798632	0.768104		6	4	0.7814870	0.806980	0.8270071
		6	0.829625	0.796964	0.766719			5.5	0.7815446	0.807086	0.8272181

Example 5. 

In this example, we have discussed the system performance measures such as $M_I$, $M_R$, $M_{WH}$, and $M_O$ under the different parameter combinations which are shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

1.

Figure 1 explains the effect of $E_O$ over ${\lambda }_1$ and γ. Whenever the arrival rate is increased, $M_O$ is increased. The $M_O$ decreases as the vacation completion rate is increased.

2.

The impact of ${\lambda }_2$ and μ over the average number of customers can be seen in Figure 2. $E_O$ increases as usually the arrival rate is increased. While increasing the service rate, the mean service time decreases, such that the $M_O$ decreases.

3.

The result of varying ${\theta }_1$ and γ over the $M_O$ is shown in Figure Table 2. When the retrial rate is increased, $M_O$ decreases. As we know, while increasing the vacation completion rate, the $M_O$ decreases due to the decrease in average time of vacation.

4.

One can identify the result of $E_O$ under the variation of ${\theta }_2$ and β from Figure 4. The $M_O$ decreases when we increase the retrial rate. Increasing the reorder rate decreases the mean reorder time. So, the customer in the orbit deceases. Thus, the average number of customers in the orbit decreases while we increase both parameters.

5.

Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 explore the impact of parameters ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2,\gamma ,\mu , and \beta$ on the $M_R,M_I$ of the system.

6.

The effect of varying the parameters ${\lambda }_1$ and γ on $M_{WH}$ is shown in Figure 12. If we increase the value of ${\lambda }_1$, the number of customer increases in the waiting hall. The vacation period decreases whenever the vacation completion rate is increased. This effect decreases the $M_{WH}$.

7.

Figure 13 explains the average number of customers present in the waiting hall over the impact of increasing ${\lambda }_2$ and μ. Due to the increase in arrivals, customers’ arrivals increase in the waiting hall. The mean service time per customer always increases as the service rate increases. So the number of customers available in the waiting hall increases.

8.

The parametric variation of ${\theta }_1$ and γ over the $M_{WH}$ is plotted in Figure 14. As we discussed earlier, $M_{WH}$ increases due to the increase in retrial rate. Increasing the parameter γ ensures the availability of the server as the vacation time decreases. So the average number of customers present in the waiting hall decreases.

Observing the figures, one can maintain and manage the necessary system performance in the business using the strategies. When a higher number of customers arrive in the system, service time and reorder time should be fast enough, which will even affect the total cost.

Figure 1. Mean number of customers in the orbit on ${\lambda }_1$ vs. $\gamma$.

Figure 1. Mean number of customers in the orbit on ${\lambda }_1$ vs. $\gamma$.

Figure 2. Mean number of customers in the orbit on${\lambda }_2$ vs. $\mu$.

Figure 2. Mean number of customers in the orbit on${\lambda }_2$ vs. $\mu$.

Figure 3. Mean number of customers in the orbit on${\theta }_1$ vs. $\gamma$.

Figure 3. Mean number of customers in the orbit on${\theta }_1$ vs. $\gamma$.

Figure 4. Mean number of customers in the orbit on${\theta }_2$ vs. $\beta$.

Figure 4. Mean number of customers in the orbit on${\theta }_2$ vs. $\beta$.

Figure 5. Mean reorder rate on${\lambda }_1$ vs. $\gamma$.

Figure 5. Mean reorder rate on${\lambda }_1$ vs. $\gamma$.

Figure 6. Mean reorder rate on${\lambda }_2$ vs. $\mu$.

Figure 6. Mean reorder rate on${\lambda }_2$ vs. $\mu$.

Figure 7. Mean reorder rate on${\theta }_1$ vs. $\gamma$.

Figure 7. Mean reorder rate on${\theta }_1$ vs. $\gamma$.

Figure 8. Mean number of items in the inventory on ${\lambda }_1$ vs. $\gamma$.

Figure 8. Mean number of items in the inventory on ${\lambda }_1$ vs. $\gamma$.

Figure 9. Mean number of items in the inventory on ${\lambda }_2$ vs. $\mu$.

Figure 9. Mean number of items in the inventory on ${\lambda }_2$ vs. $\mu$.

Figure 10. Mean number of items in the inventory on ${\theta }_1$ vs. $\gamma$.

Figure 10. Mean number of items in the inventory on ${\theta }_1$ vs. $\gamma$.

Figure 11. Mean number of items in the inventory on ${\theta }_2$ vs. $\beta$.

Figure 11. Mean number of items in the inventory on ${\theta }_2$ vs. $\beta$.

Figure 12. Mean number of customers in the waiting hall on ${\lambda }_1$ vs. $\gamma$.

Figure 12. Mean number of customers in the waiting hall on ${\lambda }_1$ vs. $\gamma$.

Figure 13. Mean number of customers in the waiting hall on ${\lambda }_2$ vs. $\mu$.

Figure 13. Mean number of customers in the waiting hall on ${\lambda }_2$ vs. $\mu$.

Figure 14. Mean number of customers in the waiting hall on ${\theta }_1$ vs. $\gamma$.

Figure 14. Mean number of customers in the waiting hall on ${\theta }_1$ vs. $\gamma$.

6. Conclusions

This stochastic model explores the single server queueing-inventory system with server multiple vacations. We assumed a vacation-dependent arrival policy for both primary and retrial customers. For the replenishment of a product, an $(s,Q)$ ordering principle is adopted whenever the current inventory level reaches the reorder point s. We applied a matrix-geometric method to analyse the steady state solutions, system performance measures, and construction of the expected total cost.

6.1. Observations

The following observations will be helpful to the readers who want to apply this idea in their retail business.

1.

The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model.

2.

The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation.

3.

The discussion of the fraction of time that the server is on vacation suggests that as the server’s vacation duration reduces, his/her fraction time also reduces.

4.

The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.

5.

The fraction of the successful rate of retrial is investigated for all the parameters. It helps us to study the retrial customers with different parameter combinations.

6.2. Limitations

When we assume ${\lambda }_1={\lambda }_2$ and ${\theta }_1={\theta }_2$, this model comes under the framework of a non-vacation-dependent arrival rate queueing-inventory system.

6.3. Managerial Implications

Running a business is not like purchasing a water bottle, where we use it one time and then throw it out. Though the initial amount we spent in the business can be calculated easily, there are many other indirect costs such as maintenance, electricity, transportation, taxes, etc. Total cost is helpful to analyse the effect of indirect costs. From the above observations, it is clear that the key features of a profitable business depend on the service rate, reorder rate, and vacation completion rate. By increasing the rate of these parameters, one can run a successful business by decreasing the total cost. A good businessman would focus more on customer satisfaction than profit. Most customers do not like to wait in the waiting hall. The results suggest that in a good business, a fast service rate must be provided. Completion of server vacation must be quick, so that those who entered into the system during the server’s vacation would obtain service, and replenishment time must be short.

In the future, we will extend this work along with the multi-server service channels and stock-dependent arrival policies of queueing-inventory systems.